David Cameron has completed his biggest reshuffle in office, demoting the trouble-making Michael Gove to chief whip, installing the Eurosceptic Philip Hammond as foreign secretary and promoting two more women to the cabinet.

The prime minister has cleared out a dozen middle-aged and older men from his ministerial ranks in order to create a more female and less privileged top team. The strategy behind the shake-up is being attributed to Lynton Crosby, Cameron's election adviser, as the party seeks to make itself look more representative of society before the election.

However, there are already signs of a backlash within the party about the scale of the sackings. Among the centrists, there is unhappiness about the departure of Dominic Grieve as attorney general, who has stood up for the European court of human rights, Ken Clarke as a cabinet minister without portfolio, who was a rare pro-European voice, and Greg Barker, an energy minister who consistently fought for climate change to remain on the agenda.

Those on the right of the Conservatives are mourning the loss of right-winger Owen Paterson as environment secretary. His departure was met with jubilation by green campaigners and dismay among the farming community.

The right was also upset at the removal of Gove – considered a radical on education policy and a hawk on foreign policy – from his role as education secretary, although the move was greeted with approval by teachers' groups with whom he has clashed.

The two new women in cabinet are Nicky Morgan, who replaces Gove as education secretary, and Liz Truss, replacing Paterson at the Department for the Environment, Food and Rural Affairs.

Other new faces include Michael Fallon, as defence secretary to replace Hammond, Jeremy Wright as the attorney general, and Stephen Crabb as Welsh secretary, instead of the sacked David Jones.

The new appointments only take the number of women in the full cabinet up to five – the same number as in 2011 – as Esther McVey, the employment minister, and Lady Stowell, the new leader of the Lords, will only have the right to attend the Tuesday morning meetings.

Stowell was initially not going to get the higher salary of her predecessor, but following outrage about the discrepancy, the Conservative party said it would top up her salary to the same as that enjoyed by Hill on the taxpayer.

Several women from the 2010 intake have been appointed to more junior ministerial roles. These are Amber Rudd as an energy minister, Priti Patel as the exchequer secretary, Penny Mordaunt as minister for coastal communities, and Claire Perry as a transport minister, while Anna Soubry was given a promotion to minister of state in the Ministry of Defence.

The choice of Hammond to replace William Hague as foreign secretary and Lord Hill of Oareford as the UK's candidate for European commissioner shows Cameron positioning trusted operators in the hope they will be able to manage his party's troubled relationship with the EU.

As the dust settled on the new appointments, a number of the changes appear already to have caused controversy and the potential for trouble.

• The government's equality policy was thrown into confusion after Cameron made Morgan the minister for women and equalities, as well as education secretary, even though she was one of a few ministers to vote against same-sex marriage. A second minister, Nick Boles, has been put in charge of implementing the rest of the gay marriage policy.

• The former defence secretary Liam Fox turned down an opportunity to return as a foreign office minister more than two years after he resigned over a controversy about his unofficial adviser Adam Werritty. The senior Tory said he was honoured to have been offered the post but, in slightly ominous comments for the prime minister, he would rather be free to speak up from the backbench on the economy, immigration and Europe.

• Labour criticised Cameron for failing to meet his promise of giving a third of ministerial appointments to women. They also pointed out that Lady Stowell will do exactly the same job as her predecessor, Lord Hill, as leader of the House of Lords, but with less seniority and £22,000 less pay.

• The newly created job of minister for coastal communities given to Penny Mordaunt, the Portsmouth North MP, appeared to be an attempt neutralise the threat of Ukip in seaside towns. Her appointment comes just weeks after the Ukip leader, Nigel Farage, hinted he may stand in the coastal constituency of Thanet South at the next election. Last year, he told the Times: "What's become really interesting is the phenomenon that no one has really noticed, which is that by accident we're becoming the seaside party."

• It emerged that Crabb once described the role of Welsh secretary as "emptied and somewhat meaningless" and raised concerns about devolution causing permanent damage to the UK. While in opposition, the Conservative MP wrote an article for Conservative Home arguing that devolution was as unsafe as "uncontrolled immigration and relentless European integration".

• Priti Patel, a rightwinger and former PR executive who has endorsed the death penalty and campaigned against plain cigarette packaging, was a controversial choice for promotion to the Treasury as exchequer secretary.

• There was confusion over whether there would be a dedicated minister for the disabled. McVey was made an employment and disabilities minister, but that appeared to be only part of her portfolio. After hours of speculation Mark Harper, previously announced as a minister of state in the Department for Work and Pensions, appeared to have his title converted to take on the role.

• Labour said all the ministers in the Department for Education had spent time at private school. Morgan went to an independent girls' school and Oxford University. Boles was educated at the independent Winchester College and Oxford University, Edward Timpson went to the private school Uppingham and to Durham University, while Lord Nash went to the independent school Milton Abbey and Oxford University. Nick Gibb, returning to his old role as an education minister, went to non-selective state, grammar and private schools, before going to Durham University.

However, late in the day, No 10 announced that the whip Sam Gyimah would join the department as an under-secretary of state. He went to state schools in the UK and Ghana before attending Oxford University.